Progress Report on One Simulated Dataset
Ying Jin
2023-09-15
I am now on the way to add the debias step to the whole algorithm. Just so that I have a set of “true parameters” for comparison, I am gonna use the simulation dataset for this part.
1 Generation mechanism
- I am gonna generate only one dataset as an example
- 500 individuals each with 1000 measures
\[\begin{aligned} Y_i(t) & \sim Bernoulli(\frac{exp(\eta_i(t))}{1+exp(\eta_i(t))}) \\ \eta_i(t) &= f_0(t)+ \xi_{i1}\sqrt{2}sin(2\pi t)+\xi_{i2}\sqrt{2}cos(2\pi t)+\xi_{i3}\sqrt{2}sin(4\pi t)+\xi_{i4}\sqrt{2}cos(4\pi t) \end{aligned}\]
where:
- \(t\) is equal-spaced on \([0, 1]\)
- \(f_0(t)=0\)
- \(\xi_k \sim N(0, \lambda_k)\), and \(\lambda_k = 1, 0.5, 0.25, 0.125\) for k = 1, 2, 3, 4 respectively.
- In fact, since there is a constant factor \(\sqrt{2}\) before each eigenfunction so that they are orthogonal
N <- 500 # sample size
J <- 1000 # number of observation points
t = seq(0,1,len=J) # observations points
# mean function
f_0 <- function(s) 0
#eigenfunctions
K <- 4 # number of eigenfunctions
phi <- sqrt(2)*cbind(sin(2*pi*t),cos(2*pi*t),
sin(4*pi*t),cos(4*pi*t))
# eigenvalues
lambda = 0.5^(0:(K-1)) # generated training data
## score
xi <- matrix(rnorm(N*K),N,K)
xi <- xi %*% diag(sqrt(lambda))
# subject-specific random effect
b_i <- xi %*% t(phi); # of size N by J
# latent gaussian function
eta_i <- t(vapply(1:N, function(x){
f_0(t) + b_i[x,]
}, numeric(J)))
# outcome binary function
Y_i <- matrix(rbinom(N*J, size=1, prob=plogis(eta_i)),
N, J, byrow=FALSE)
# format into dataframe
# id = subject identifier (factor variable)
# sind = numeric value corresponding to the observed functional domain
# Y = functional response (binary)
# sind_inx = numeric value associated with order of "sind"
# this is not necessary, but may be of convenience when you implement the method
train_df <- data.frame(id = factor(rep(1:N, each=J)),
t = rep(t, N),
Y = as.vector(t(Y_i)),
eta_i = as.vector(t(eta_i)),
sind = rep(1:J, N))
# visualization
rand_id <- sample(N, size = 4)
train_df %>% filter(id %in% rand_id) %>%
ggplot()+
geom_point(aes(x=sind, y=Y), size = 0.5)+
geom_line(aes(x=sind, y=plogis(eta_i)), col = "red")+
#geom_line(aes(x=sind_inx, y=eta_i), col = "blue")+
facet_wrap(~id)
2 fGFPCA model fitting
- Training sample: N=500
2.1 Bin data
- Bin every 10 consecutive observations
# bin data
bin_w <- 10 # bin width
n_bin <- J/bin_w # number of bins
brks <- seq(0, J, by = bin_w) # cutoff points
mid <- (brks+bin_w/2)[1:n_bin] # mid points
train_df$bin <- cut(train_df$sind, breaks = brks, include.lowest = T, labels = mid)
train_df$bin <- as.numeric(as.character(train_df$bin))
# unique(df$bin)
train_df %>%
filter(id %in% rand_id) %>%
group_by(id, bin) %>%
summarise(num = sum(Y)) %>%
ggplot()+
geom_point(aes(x=bin, y=num), size = 0.5)+
facet_wrap(~id)+
labs(x="Time", y = "Activity", title = "Number of active nimutes within each bin")
2.2 Local GLMM
- Used glmer with PIRLS (nAGQ=0) for local GLMMs step
- Fit on the training set (N=500)
- Use original time index ([0, 1]) as independent variable
# fit model on the training set
train_bin_lst <- split(train_df, f = train_df$bin)
# length(train_bin_lst)
# lapply(train_bin_lst, dim)
# local GLMM and estimate latent function
# use PIRLS (nAGQ=0) to avoid near-unidentifiability issues
t1=Sys.time()
df_est_latent <- lapply(train_bin_lst, function(x){pred_latent(x, n_node = 0)})
t2= Sys.time()
t_local_glmm <- t2-t1 # less than 5 secondsdf_est_latent <- bind_rows(df_est_latent)
# head(df_est_latent)
# example estimated latent function
train_id <- unique(train_df$id)
rand_id <- sample(train_id, 4)
df_est_latent %>%
filter(id %in% rand_id) %>%
mutate(eta_hat = exp(eta_hat)/(1+exp(eta_hat))) %>%
mutate(eta_i = exp(eta_i)/(1+exp(eta_i))) %>%
ggplot()+
geom_line(aes(x=t, y=eta_hat, group = id, col = "estimated"))+
geom_line(aes(x=t, y=eta_i, group = id, col = "true"))+
geom_point(aes(x=t, y = Y, group = id), size = 0.5)+
facet_wrap(~id, scales = "free")+
labs(x = "Time", y = "Estimated latent function (probablity scale)")
2.3 FPCA
uni_eta_hat <- df_est_latent %>% filter(bin==sind)
mid_t <- df_est_latent$t[df_est_latent$bin==df_est_latent$sind]
mid_t <- unique(mid_t) # the time points correspoinding to bin midpoints
mat_est_unique <- matrix(uni_eta_hat$eta_hat,
nrow=length(train_id),
ncol=n_bin, byrow = F)
# row index subject, column binned time
# dim(mat_est_unique)
t1 <- Sys.time()
fpca_mod <- fpca.face(mat_est_unique, argvals = mid_t, var=T)
t2 <- Sys.time()
t_fpca <- t2-t1 # less than one second- Estimated mean
plot(mid_t, fpca_mod$mu, type = "l", xlab = "bin", ylab = "Mean")
- Estimated eigenfunctions
par(mfrow=c(2,2))
plot(mid_t, fpca_mod$efunctions[, 1], type="l", xlab="bin", ylab="PC1")
plot(mid_t, fpca_mod$efunctions[, 2], type="l", xlab="bin", ylab="PC2")
plot(mid_t, fpca_mod$efunctions[, 3], type="l", xlab="bin", ylab="PC3")
plot(mid_t, fpca_mod$efunctions[, 4], type="l", xlab="bin", ylab="PC4")
- Correlation matrix
# plot correlation matrix
heatmap(cov2cor(fpca_mod$VarMats[[1]]), Rowv = NA, Colv = NA, main = "Correlation")
- Estimated eigenvalues
# estimated eigenvalues from FPCA (biased)
fpca_mod$evalues[1:4]## [1] 50.086753 23.167732 11.406167 6.0594152.4 Debias (step 4)
According to the fGFPCA paper, in this step we do two things:
- Project the eigenfunctions with B-spline basis back to the original grid
- Debias the eigenvalues with GLMM (mgcv::bam, mgcm::gamm or gamm4:gamm) to be used for Laplace approximation
2.4.1 Porjection with B-spline basis
- Use fastGFPCA::reeval_efunctions. Cannot install the package so used the source code directly.
- I would like to use time points on the [0, 1] scale as evaluation points when creating basis functions.
- Also, take number of knots and order from the FPCA model in step 3 (knots = 35, cubic spline basis).
- We don’t project the mean function here. Instead, we will use the re-evaluated mean function from the next part (bam model used for debias)
# order of b splines
p <- 3
# knots
knots <- 35
knots_values <- seq(-p, knots + p, length = knots + 1 + 2 *p)/knots
knots_values <- knots_values * (max(mid_t) - min(mid_t)) + min(mid_t)
# evaluate B-splines on binned grid
B <- spline.des(knots = knots_values, x = mid_t, ord = p + 1,
outer.ok = TRUE)$design
# evaluate B-splines on original grid
Bnew <- spline.des(knots = knots_values, x = unique(train_df$t), ord = p + 1,
outer.ok = TRUE)$design
# project binned eigenfunctions onto the original grid
efunctions_new <- matrix(NA, J, K)
for(k in 1:K){
lm_mod <- lm(fpca_mod$efunctions[,k] ~ B-1)
efunctions_new[,k] <- Bnew %*% coef(lm_mod)
}
# project mean functions
# lm_mod <- lm(fpca_mod$mu ~ B-1)
# mu_new <- Bnew %*% coef(lm_mod)# visualization
par(mfrow=c(2,2))
plot(mid_t, fpca_mod$efunctions[, 1], xlab="bin", ylab="PC1", pch=20, cex = 0.5)
lines(t, efunctions_new[, 1], col="blue")
plot(mid_t, fpca_mod$efunctions[, 2], xlab="bin", ylab="PC2", pch=20, cex = 0.5)
lines(t, efunctions_new[, 2], col="blue")
plot(mid_t, fpca_mod$efunctions[, 3], xlab="bin", ylab="PC3", pch=20, cex = 0.5)
lines(t, efunctions_new[, 3], col="blue")
plot(mid_t, fpca_mod$efunctions[, 4], xlab="bin", ylab="PC4", pch=20, cex = 0.5)
lines(t, efunctions_new[, 4], col="blue")
2.4.2 Debias with eigenvalues
- The bias was caused by the misspecification of latent process (assume constant effect over smooth function).
- What needs re-evaluation are eigenvalues (variance estimates) and the mean function. They will be used in Laplace Approximation for out-of-sample prediction.
- We do not need to debias individual scores because the scores for training sample will not be used for out-of-sample prediction. And out-of-sample scores are not estimated by FPCA but Laplace approximation.
- I will use the projected eigenfunctions from above.
- I am using mgcv::bam and setting method = “fREML” and discrete = TRUE to speed up computation
# dim(fpca_mod$efunctions)
df_phi <- data.frame(t = t, efunctions_new)
colnames(df_phi) <- c("t", paste0("phi", 1:4))
# train_df$bin <- as.numeric(as.character(train_df$bin))
train_df <- train_df %>% left_join(df_phi, by = "t")
train_df$id <- as.factor(train_df$id)# usethis::edit_r_environ()
t1 <- Sys.time()
debias_glmm <- bam(Y ~ s(t, bs="cc", k=10)+
s(id, by=phi1, bs="re")+
s(id, by=phi2, bs="re")+
s(id, by=phi3, bs="re")+
s(id, by=phi4, bs="re"),
family = binomial, data=train_df,
method = "fREML",
discrete = TRUE)
t2 <- Sys.time()
t_debias <- t2-t1 # less than 45 secondsWe will need the mean function from this model, derived below:
The shape of the function depends very much on how the basis functions are constructed. For example, cubic spline would lead to a linear mean and cyclic cubic spline would lead to the function below. What’s a good way to decided?
new_mu <- predict(debias_glmm, type = "terms")
length(unique(new_mu[,1]))## [1] 1000new_mu <- unique(new_mu[,1])
plot(t, new_mu, type = "l")
And now I try to extract the variance of scores (debiased eigenvalues), using the conclusion that connects smoothing parameters to variance
\[\hat{p}_k=1/\hat{\lambda}_k^2\]
# transformed smoothing parameters
new_lambda <- 1/debias_glmm$sp[2:5]
new_lambda## s(id):phi1 s(id):phi2 s(id):phi3 s(id):phi4
## 95.32322 47.95213 25.73439 13.18567# new_lambda/lambda
# new_lambda/fpca_mod$evalues[1:4]This eigenvalues, as well as the eigenfunctions, are actually not on the same scale of true values. By looking into the fpca.face source code, eigenfunctions are scaled by the square root of the number of total observations points. Since our model was fitted on the binnd grid, the actual eigenvaleus and eigenfunctions would be:
\[\begin{aligned} \phi_k(t) & = \sqrt{n_{bin}}\hat{\phi}_k(t) \\ \lambda_k & = \hat{\lambda}_k/n_{bin} \end{aligned}\]
Let’s rescale these quatities as below
- Re-scaled eigenvalues
scaled_lambda <- new_lambda/n_bin- Re-scaled eigenfunctions
efunctions_scaled <- efunctions_new*sqrt(n_bin)
par(mfrow=c(2,2))
plot(t, efunctions_scaled[, 1], type="l", xlab="bin", ylab="PC1")
plot(t, efunctions_scaled[, 2], type="l", xlab="bin", ylab="PC2")
plot(t, efunctions_scaled[, 3], type="l", xlab="bin", ylab="PC3")
plot(t, efunctions_scaled[, 4], type="l", xlab="bin", ylab="PC4")
- Both much closer to their true values
3 Out-of-sample prediction with Laplace approximation
I will now estimate individual score of test sample using Laplace Approximation, and calculated the predicted track.
- Generate additional 200 subjects for out-of-sample performance evaluation
- Maximum observations time: 0.2, 0.4, 0.6, 0.8
- Prediction window: 0.2-0.4, 0.4-0.6, 0.6-0.8, 0.8-1.0
3.1 Generate test data
# generated testing data
N_test <- 200
b_test <- matrix(rnorm(N_test*K), N_test, K) %*% diag(sqrt(lambda)) %*% t(phi) # of size N by J
# dim(b_test)
# latent gaussian function
eta_test <- t(vapply(1:N_test, function(x){
f_0(x) + b_i[x,]
}, numeric(J)))
# outcome binary function
Y_test <- matrix(rbinom(N_test*J, size=1, prob=plogis(eta_test)),
N_test, J, byrow=FALSE)
# format into dataframe
test_df <- data.frame(id = factor(rep(1:N_test, each=J)),
t = rep(t, N_test),
Y = as.vector(t(Y_test)),
eta_i = as.vector(t(eta_test)),
sind = rep(1:J, N_test))
# bin
test_df$bin <- cut(test_df$sind, breaks = brks, include.lowest = T, labels = mid)
test_df$bin <- as.numeric(as.character(test_df$bin))
test_df$id <- as.factor(test_df$id)
test_id <- unique(test_df$id)3.2 Laplace approximation
# the model
## now, with expanded mean and eigenfuncitons, a Bernoulli distribution at each time point
Model <- function(parm, Data){
xi <- parm[Data$pos.xi]
# log-prior
xi.prior <- dmvnorm(xi, mean = rep(0, Data$K), sigma=Data$tao, log = TRUE)
# log-posterior likelihood
eta <- Data$f0+Data$X %*% xi
p <- exp(eta)/(1+exp(eta))
LL <- sum(dbern(x=Data$y, prob=p, log = TRUE)) # log likelihood of Y|xi
LP <- LL+sum(xi.prior) # joint log likelihood of (Y, xi)
# output
Modelout <- list(LP=LP, Dev=-2*LL, Monitor=LL,
yhat=Data$y, parm=parm)
return(Modelout)
}# parameter names
name_lst <- as.list(rep(0, K))
names(name_lst) <- paste("xi", 1:K, sep = "")
parm.names <- as.parm.names(name_lst)
pos.xi <- grep("xi", parm.names)
PGF <- function(Data) {
xi <- rnorm(Data$K)
return(xi)
}
mon.names <- "LP"# approximation process
# container
pred_list <- list()
# score <- array(NA, dim=c(length(test_id), K, 4))
# dimensions: subject, eigenfunction, maximum observation time
# prediction for a single subject
t1 <- Sys.time()
# per subject
for(i in seq_along(test_id)){
df_i <- test_df %>% filter(id==test_id[i])
# per max obs time
for(tmax in c(0.2, 0.4, 0.6, 0.8)){
df_it <- df_i %>% filter(t <= tmax)
max_rid <- nrow(df_it)
# into a list
MyData <- list(K=K, PGF=PGF,
X=efunctions_new[1:max_rid, ],
mon.names=mon.names,
parm.names=parm.names,
pos.xi=pos.xi,
y=df_it$Y,
tao=diag(new_lambda), f0=new_mu[1:max_rid])
# fit laplace approximation
Fit <- LaplaceApproximation(Model, parm = rep(0, K), Data=MyData, Method = "BFGS")
# Fit <- LaplacesDemon(Model, Data=MyData, Initial.Values = parm = rep(0, K), Method = "BFGS")
score <- Fit$Summary1[, "Mode"]
# prediction
eta_pred_out <- new_mu+efunctions_new%*%score
df_i[ , paste0("pred", tmax)] <- eta_pred_out[,1]
}
# df_i$pred0.2[df_i$t<=0.2] <- NA
# df_i$pred0.4[df_i$t<=0.4] <- NA
# df_i$pred0.6[df_i$t<=0.6] <- NA
# df_i$pred0.8[df_i$t<=0.8] <- NA
pred_list[[i]] <- df_i
}
t2 <- Sys.time()
t_pred <- t2-t1 # About 3.5 minutesdf_pred <- bind_rows(pred_list)
df_pred$pred0.2[df_pred$t<=0.2] <- NA
df_pred$pred0.4[df_pred$t<=0.4] <- NA
df_pred$pred0.6[df_pred$t<=0.6] <- NA
df_pred$pred0.8[df_pred$t<=0.8] <- NA
rand_test_id <- sample(test_id, 4)
p1<-df_pred %>%
filter(id %in% rand_test_id) %>%
mutate_at(vars(eta_i, pred0.2, pred0.4, pred0.6, pred0.8), function(x){exp(x)/(1+exp(x))}) %>%
ggplot()+
geom_point(aes(x=t, y=Y), size = 0.2)+
geom_line(aes(x=t, y=eta_i, col = "True"))+
geom_line(aes(x=t, y=pred0.2, col = "0.2"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.4, col = "0.4"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.6, col = "0.6"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.8, col = "0.8"), linetype="dashed", na.rm=T)+
facet_wrap(~id)+
labs(title = "Unscaled, debiased eigenvalues")## With debiased eigenvalues, also scale eigenvaleus and eigenfuncitons
# container
pred_list3 <- list()
# per subject
for(i in seq_along(test_id)){
df_i <- test_df %>% filter(id==test_id[i])
# per max obs time
for(tmax in c(0.2, 0.4, 0.6, 0.8)){
df_it <- df_i %>% filter(t <= tmax)
max_rid <- nrow(df_it)
# into a list
MyData <- list(K=K, PGF=PGF,
X=efunctions_scaled[1: max_rid, ],
mon.names=mon.names,
parm.names=parm.names,
pos.xi=pos.xi,
y=df_it$Y,
tao=diag(scaled_lambda), f0=new_mu[1:max_rid])
# fit laplace approximation
Fit <- LaplaceApproximation(Model, parm = rep(0, K), Data=MyData, Method = "BFGS")
score <- Fit$Summary1[, "Mode"]
# prediction
eta_pred_out <- new_mu+efunctions_scaled%*%score
df_i[ , paste0("pred", tmax)] <- eta_pred_out[,1]
}
# df_i$pred0.2[df_i$t<=0.2] <- NA
# df_i$pred0.4[df_i$t<=0.4] <- NA
# df_i$pred0.6[df_i$t<=0.6] <- NA
# df_i$pred0.8[df_i$t<=0.8] <- NA
pred_list3[[i]] <- df_i
}df_pred3 <- bind_rows(pred_list3)
df_pred3$pred0.2[df_pred3$t<=0.2] <- NA
df_pred3$pred0.4[df_pred3$t<=0.4] <- NA
df_pred3$pred0.6[df_pred3$t<=0.6] <- NA
df_pred3$pred0.8[df_pred3$t<=0.8] <- NA
# overview of predicted track
p3 <- df_pred3 %>%
filter(id %in% rand_test_id) %>%
mutate_at(vars(eta_i, pred0.2, pred0.4, pred0.6, pred0.8), function(x){exp(x)/(1+exp(x))}) %>%
ggplot()+
geom_point(aes(x=t, y=Y), size = 0.2)+
geom_line(aes(x=t, y=eta_i, col = "True"))+
geom_line(aes(x=t, y=pred0.2, col = "0.2"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.4, col = "0.4"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.6, col = "0.6"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred0.8, col = "0.8"), linetype="dashed", na.rm=T)+
facet_wrap(~id)+
labs(title = "Scaled, unbiased eigenvalues")t_1sim <- (t_local_glmm+t_fpca+t_debias+t_pred)I would like to see if the following operations can improve predictive performance:
- Debiasing eigenvalues from FPCA
- Scale the eigenfunctions and debiased eigenvalues (from mgcv::bam) so that they are close to the true values for data generation
ggarrange(p1, p3, nrow = 1, common.legend = T)
- Time spent on one iteration: 193.82 seconds
- Debiasing eigenvalues helped with prediction. Scaling eigenfunctiosn and eigenvalues, on the other hand, ended up being more complicated.
3.3 ISE
# calcualte ISE
err1 <- df_pred %>%
mutate(err1 = (pred0.2-eta_i)^2,
err2 = (pred0.4-eta_i)^2,
err3 = (pred0.6-eta_i)^2,
err4 = (pred0.8-eta_i)^2) %>%
select(id, t, starts_with("err"))
err1$window = cut(err1$t, breaks = seq(0, 1, by = 0.2), include.lowest = T)
# table(err1$window)
err1 <- split(err1, f = err1$window)
tb1 <- lapply(err1, function(x){
x %>% group_by(id) %>%
summarise_at(vars(err1, err2, err3, err4), sum) %>%
summarise_at(vars(err1, err2, err3, err4), mean)
}) %>% bind_rows(.id = "Window")# calcualte ISE
err3 <- df_pred3 %>%
mutate(err1 = (pred0.2-eta_i)^2,
err2 = (pred0.4-eta_i)^2,
err3 = (pred0.6-eta_i)^2,
err4 = (pred0.8-eta_i)^2) %>%
select(id, t, starts_with("err"))
err3$window = cut(err3$t, breaks = seq(0, 1, by = 0.2), include.lowest = T)
# table(err1$window)
err3 <- split(err3, f = err3$window)
tb3 <- lapply(err3, function(x){
x %>% group_by(id) %>%
summarise_at(vars(err1, err2, err3, err4), sum) %>%
summarise_at(vars(err1, err2, err3, err4), mean)
}) %>% bind_rows(.id = "Window")# calcualte ISE
options(knitr.kable.NA = "")
ise <- full_join(tb1, tb3, by = "Window")
colnames(ise) <-c("Window", rep(seq(0.2, 0.8, by =0.2), 2))
ise %>%
kable(digits = 3, table.attr = "style = \"color: black;\"") %>%
kable_styling(full_width = F) %>%
add_header_above(c(" "= 1, "Unscaled" = 4, "Scaled" = 4)) %>%
add_header_above(c(" "=1, "Maximum observation time" = 8))| Window | 0.2 | 0.4 | 0.6 | 0.8 | 0.2 | 0.4 | 0.6 | 0.8 |
|---|---|---|---|---|---|---|---|---|
| [0,0.2] | ||||||||
| (0.2,0.4] | 150.474 | 150.251 | ||||||
| (0.4,0.6] | 192.605 | 76.254 | 193.101 | 81.244 | ||||
| (0.6,0.8] | 224.855 | 53.395 | 18.161 | 224.123 | 56.692 | 18.338 | ||
| (0.8,1] | 118.223 | 88.065 | 22.398 | 13.259 | 118.133 | 95.631 | 22.768 | 13.318 |
- Looks like the debiased eigenvalues helped.
- Scaling, on the other hand, doesn’t seem to help in general. But shouldn’t scaling improve performance?
- Also, it seems like the longer the observed track, the less different the performance is.
3.4 AUC
# calcualte AUC
auc_df1 <- df_pred %>%
mutate(window = cut(t, breaks = seq(0, 1, by = 0.2), include.lowest = T)) %>%
select(Y, starts_with("pred"), window)
obs_track <- seq(0.2, 0.8, by = 0.2)
pred_window <- unique(auc_df1$window)
auc_mat <- matrix(NA, length(pred_window), length(obs_track))
for(j in seq_along(obs_track)){
for(i in (j+1):5){
this_Y <- auc_df1$Y[auc_df1$window==pred_window[i]]
this_pred <- auc_df1[auc_df1$window==pred_window[i],
paste0("pred", obs_track[j])]
this_perf <- performance(prediction(this_pred, this_Y), measure = "auc")
auc_mat[i ,j] <- this_perf@y.values[[1]]
}
}# calcualte AUC
auc_df3 <- df_pred3 %>%
mutate(window = cut(t, breaks = seq(0, 1, by = 0.2), include.lowest = T)) %>%
select(Y, starts_with("pred"), window)
auc_mat3 <- matrix(NA, length(pred_window), length(obs_track))
for(j in seq_along(obs_track)){
for(i in (j+1):5){
this_Y <- auc_df3$Y[auc_df3$window==pred_window[i]]
this_pred <- auc_df3[auc_df3$window==pred_window[i],
paste0("pred", obs_track[j])]
this_perf <- performance(prediction(this_pred, this_Y), measure = "auc")
auc_mat3[i ,j] <- this_perf@y.values[[1]]
}
}# calcualte AUC
options(knitr.kable.NA = "")
auc <- cbind(auc_mat, auc_mat3)
rownames(auc) <- unique(auc_df1$window)
colnames(auc) <-c(rep(seq(0.2, 0.8, by = 0.2), 2))
auc %>%
kable(digits = 3, table.attr = "style = \"color: black;\"") %>%
kable_styling(full_width = F) %>%
add_header_above(c(" "= 1, "Unscaled" = 4, "Scaled" = 4)) %>%
add_header_above(c(" "=1, "Maximum observation time" = 8))| 0.2 | 0.4 | 0.6 | 0.8 | 0.2 | 0.4 | 0.6 | 0.8 | |
|---|---|---|---|---|---|---|---|---|
| [0,0.2] | ||||||||
| (0.2,0.4] | 0.737 | 0.737 | ||||||
| (0.4,0.6] | 0.666 | 0.744 | 0.666 | 0.743 | ||||
| (0.6,0.8] | 0.701 | 0.782 | 0.795 | 0.701 | 0.782 | 0.795 | ||
| (0.8,1] | 0.753 | 0.768 | 0.797 | 0.8 | 0.753 | 0.765 | 0.797 | 0.8 |
3.5 Investigate a bit more on the difference in prediction
- The scaled and unscaled parameters seem to make a difference in prediction, which they shouldn’t
par(mfrow=c(2, 2))
plot(df_pred$pred0.2, df_pred3$pred0.2, pch=20, cex = 0.5,
xlab="unscaled", ylab="scaled", main = "Up to 0.2")
plot(df_pred$pred0.4, df_pred3$pred0.4, pch=20, cex = 0.5,
xlab="unscaled", ylab="scaled", main = "Up to 0.4")
plot(df_pred$pred0.6, df_pred3$pred0.6, pch=20, cex = 0.5,
xlab="unscaled", ylab="scaled", main = "Up to 0.6")
plot(df_pred$pred0.8, df_pred3$pred0.8, pch=20, cex = 0.5,
xlab="unscaled", ylab="scaled", main = "Up to 0.8")
data.frame(
df_pred$pred0.4[df_pred$t>0.4],
df_pred3$pred0.4[df_pred3$t>0.4]
)## df_pred.pred0.4.df_pred.t...0.4. df_pred3.pred0.4.df_pred3.t...0.4.
## 1 -1.390934e+00 -1.337147e+00
## 2 -1.380528e+00 -1.327535e+00
## 3 -1.370120e+00 -1.317923e+00
## 4 -1.359710e+00 -1.308312e+00
## 5 -1.349298e+00 -1.298701e+00
## 6 -1.338887e+00 -1.289093e+00
## 7 -1.328476e+00 -1.279488e+00
## 8 -1.318068e+00 -1.269887e+00
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## 1552 1.462757e+00 1.280489e+00
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## 1579 1.557062e+00 1.387093e+00
## 1580 1.559768e+00 1.390423e+00
## 1581 1.562409e+00 1.393699e+00
## 1582 1.564984e+00 1.396921e+00
## 1583 1.567493e+00 1.400087e+00
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## 1590 1.583118e+00 1.420619e+00
## 1591 1.585063e+00 1.423307e+00
## 1592 1.586933e+00 1.425932e+00
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## 1595 1.592088e+00 1.433409e+00
## 1596 1.593652e+00 1.435767e+00
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## 1599 1.597866e+00 1.442424e+00
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## 1606 1.604834e+00 1.455421e+00
## 1607 1.605493e+00 1.456977e+00
## 1608 1.606065e+00 1.458456e+00
## 1609 1.606551e+00 1.459857e+00
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## 1624 1.603011e+00 1.470996e+00
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## 1631 1.594049e+00 1.469433e+00
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## 1633 1.590603e+00 1.468156e+00
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## 1644 1.564500e+00 1.454378e+00
## 1645 1.561522e+00 1.452549e+00
## 1646 1.558441e+00 1.450623e+00
## 1647 1.555258e+00 1.448598e+00
## 1648 1.551973e+00 1.446476e+00
## 1649 1.548585e+00 1.444255e+00
## 1650 1.545095e+00 1.441935e+00
## 1651 1.541501e+00 1.439515e+00
## 1652 1.537804e+00 1.436997e+00
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## 1655 1.526092e+00 1.428840e+00
## 1656 1.521980e+00 1.425920e+00
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## 1658 1.513447e+00 1.419780e+00
## 1659 1.509026e+00 1.416560e+00
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## 1661 1.499874e+00 1.409818e+00
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## 1671 1.447965e+00 1.370105e+00
## 1672 1.442213e+00 1.365584e+00
## 1673 1.436361e+00 1.360963e+00
## 1674 1.430407e+00 1.356243e+00
## 1675 1.424352e+00 1.351423e+00
## 1676 1.418196e+00 1.346503e+00
## 1677 1.411939e+00 1.341483e+00
## 1678 1.405582e+00 1.336364e+00
## 1679 1.399124e+00 1.331145e+00
## 1680 1.392566e+00 1.325827e+00
## 1681 1.385907e+00 1.320409e+00
## 1682 1.379148e+00 1.314891e+00
## 1683 1.372290e+00 1.309274e+00
## 1684 1.365332e+00 1.303559e+00
## 1685 1.358275e+00 1.297745e+00
## 1686 1.351122e+00 1.291835e+00
## 1687 1.343871e+00 1.285827e+00
## 1688 1.336524e+00 1.279724e+00
## 1689 1.329081e+00 1.273525e+00
## 1690 1.321544e+00 1.267232e+00
## 1691 1.313913e+00 1.260845e+00
## 1692 1.306189e+00 1.254365e+00
## 1693 1.298373e+00 1.247792e+00
## 1694 1.290465e+00 1.241127e+00
## 1695 1.282466e+00 1.234372e+00
## 1696 1.274376e+00 1.227525e+00
## 1697 1.266198e+00 1.220589e+00
## 1698 1.257931e+00 1.213564e+00
## 1699 1.249576e+00 1.206451e+00
## 1700 1.241133e+00 1.199250e+00
## 1701 1.232605e+00 1.191962e+00
## 1702 1.223990e+00 1.184587e+00
## 1703 1.215291e+00 1.177127e+00
## 1704 1.206508e+00 1.169582e+00
## 1705 1.197642e+00 1.161953e+00
## 1706 1.188692e+00 1.154240e+00
## 1707 1.179661e+00 1.146444e+00
## 1708 1.170549e+00 1.138566e+00
## 1709 1.161356e+00 1.130606e+00
## 1710 1.152084e+00 1.122566e+00
## 1711 1.142733e+00 1.114445e+00
## 1712 1.133305e+00 1.106246e+00
## 1713 1.123799e+00 1.097968e+00
## 1714 1.114218e+00 1.089613e+00
## 1715 1.104563e+00 1.081183e+00
## 1716 1.094834e+00 1.072678e+00
## 1717 1.085033e+00 1.064099e+00
## 1718 1.075161e+00 1.055448e+00
## 1719 1.065218e+00 1.046725e+00
## 1720 1.055207e+00 1.037932e+00
## 1721 1.045128e+00 1.029069e+00
## 1722 1.034983e+00 1.020139e+00
## 1723 1.024772e+00 1.011141e+00
## 1724 1.014496e+00 1.002077e+00
## 1725 1.004157e+00 9.929489e-01
## 1726 9.937560e-01 9.837566e-01
## 1727 9.832937e-01 9.745015e-01
## 1728 9.727714e-01 9.651849e-01
## 1729 9.621902e-01 9.558077e-01
## 1730 9.515512e-01 9.463711e-01
## 1731 9.408556e-01 9.368762e-01
## 1732 9.301043e-01 9.273241e-01
## 1733 9.192987e-01 9.177158e-01
## 1734 9.084396e-01 9.080525e-01
## 1735 8.975284e-01 8.983352e-01
## 1736 8.865660e-01 8.885651e-01
## 1737 8.755536e-01 8.787432e-01
## 1738 8.644923e-01 8.688706e-01
## 1739 8.533833e-01 8.589485e-01
## 1740 8.422275e-01 8.489779e-01
## 1741 8.310260e-01 8.389598e-01
## 1742 8.197799e-01 8.288954e-01
## 1743 8.084903e-01 8.187857e-01
## 1744 7.971581e-01 8.086318e-01
## 1745 7.857844e-01 7.984348e-01
## 1746 7.743702e-01 7.881957e-01
## 1747 7.629166e-01 7.779156e-01
## 1748 7.514247e-01 7.675955e-01
## 1749 7.398954e-01 7.572366e-01
## 1750 7.283298e-01 7.468399e-01
## 1751 7.167290e-01 7.364065e-01
## 1752 7.050940e-01 7.259374e-01
## 1753 6.934258e-01 7.154337e-01
## 1754 6.817255e-01 7.048965e-01
## 1755 6.699941e-01 6.943269e-01
## 1756 6.582327e-01 6.837259e-01
## 1757 6.464423e-01 6.730946e-01
## 1758 6.346240e-01 6.624341e-01
## 1759 6.227788e-01 6.517454e-01
## 1760 6.109077e-01 6.410296e-01
## 1761 5.990118e-01 6.302878e-01
## 1762 5.870921e-01 6.195211e-01
## 1763 5.751496e-01 6.087304e-01
## 1764 5.631855e-01 5.979170e-01
## 1765 5.512008e-01 5.870818e-01
## 1766 5.391964e-01 5.762260e-01
## 1767 5.271735e-01 5.653505e-01
## 1768 5.151328e-01 5.544563e-01
## 1769 5.030751e-01 5.435441e-01
## 1770 4.910008e-01 5.326143e-01
## 1771 4.789104e-01 5.216676e-01
## 1772 4.668046e-01 5.107045e-01
## 1773 4.546839e-01 4.997256e-01
## 1774 4.425487e-01 4.887315e-01
## 1775 4.303997e-01 4.777227e-01
## 1776 4.182374e-01 4.666999e-01
## 1777 4.060624e-01 4.556636e-01
## 1778 3.938751e-01 4.446144e-01
## 1779 3.816762e-01 4.335528e-01
## 1780 3.694661e-01 4.224795e-01
## 1781 3.572455e-01 4.113950e-01
## 1782 3.450149e-01 4.002998e-01
## 1783 3.327747e-01 3.891947e-01
## 1784 3.205256e-01 3.780800e-01
## 1785 3.082682e-01 3.669565e-01
## 1786 2.960029e-01 3.558247e-01
## 1787 2.837302e-01 3.446851e-01
## 1788 2.714509e-01 3.335384e-01
## 1789 2.591653e-01 3.223851e-01
## 1790 2.468740e-01 3.112258e-01
## 1791 2.345776e-01 3.000611e-01
## 1792 2.222767e-01 2.888915e-01
## 1793 2.099717e-01 2.777176e-01
## 1794 1.976632e-01 2.665400e-01
## 1795 1.853518e-01 2.553593e-01
## 1796 1.730417e-01 2.441787e-01
## 1797 1.607519e-01 2.330125e-01
## 1798 1.485052e-01 2.218773e-01
## 1799 1.363244e-01 2.107901e-01
## 1800 1.242322e-01 1.997677e-01
## 1801 -6.506134e-02 -6.503087e-02
## 1802 -5.826280e-02 -5.823211e-02
## 1803 -5.142213e-02 -5.139124e-02
## 1804 -4.453961e-02 -4.450851e-02
## 1805 -3.761549e-02 -3.758418e-02
## 1806 -3.065003e-02 -3.061852e-02
## 1807 -2.364348e-02 -2.361177e-02
## 1808 -1.659611e-02 -1.656421e-02
## 1809 -9.508168e-03 -9.476077e-03
## 1810 -2.379919e-03 -2.347642e-03
## 1811 4.788382e-03 4.820841e-03
## 1812 1.199647e-02 1.202911e-02
## 1813 1.924410e-02 1.927691e-02
## 1814 2.653101e-02 2.656399e-02
## 1815 3.385693e-02 3.389008e-02
## 1816 4.122161e-02 4.125492e-02
## 1817 4.862479e-02 4.865827e-02
## 1818 5.606622e-02 5.609985e-02
## 1819 6.354563e-02 6.357941e-02
## 1820 7.106277e-02 7.109670e-02
## 1821 7.861738e-02 7.865145e-02
## 1822 8.620920e-02 8.624341e-02
## 1823 9.383797e-02 9.387231e-02
## 1824 1.015034e-01 1.015379e-01
## 1825 1.092053e-01 1.092399e-01
## 1826 1.169434e-01 1.169781e-01
## 1827 1.247174e-01 1.247523e-01
## 1828 1.325271e-01 1.325620e-01
## 1829 1.403721e-01 1.404072e-01
## 1830 1.482523e-01 1.482875e-01
## 1831 1.561672e-01 1.562025e-01
## 1832 1.641164e-01 1.641517e-01
## 1833 1.720992e-01 1.721347e-01
## 1834 1.801154e-01 1.801509e-01
## 1835 1.881642e-01 1.881998e-01
## 1836 1.962453e-01 1.962810e-01
## 1837 2.043582e-01 2.043940e-01
## 1838 2.125023e-01 2.125382e-01
## 1839 2.206773e-01 2.207132e-01
## 1840 2.288825e-01 2.289185e-01
## 1841 2.371175e-01 2.371535e-01
## 1842 2.453818e-01 2.454179e-01
## 1843 2.536749e-01 2.537110e-01
## 1844 2.619963e-01 2.620325e-01
## 1845 2.703455e-01 2.703817e-01
## 1846 2.787221e-01 2.787583e-01
## 1847 2.871255e-01 2.871618e-01
## 1848 2.955552e-01 2.955915e-01
## 1849 3.040108e-01 3.040471e-01
## 1850 3.124917e-01 3.125281e-01
## 1851 3.209975e-01 3.210339e-01
## 1852 3.295277e-01 3.295641e-01
## 1853 3.380818e-01 3.381182e-01
## 1854 3.466592e-01 3.466956e-01
## 1855 3.552595e-01 3.552959e-01
## 1856 3.638823e-01 3.639187e-01
## 1857 3.725269e-01 3.725633e-01
## 1858 3.811930e-01 3.812294e-01
## 1859 3.898799e-01 3.899162e-01
## 1860 3.985870e-01 3.986233e-01
## 1861 4.073135e-01 4.073497e-01
## 1862 4.160588e-01 4.160950e-01
## 1863 4.248221e-01 4.248583e-01
## 1864 4.336029e-01 4.336391e-01
## 1865 4.424004e-01 4.424365e-01
## 1866 4.512139e-01 4.512499e-01
## 1867 4.600427e-01 4.600787e-01
## 1868 4.688862e-01 4.689221e-01
## 1869 4.777436e-01 4.777795e-01
## 1870 4.866143e-01 4.866501e-01
## 1871 4.954976e-01 4.955333e-01
## 1872 5.043928e-01 5.044284e-01
## 1873 5.132992e-01 5.133347e-01
## 1874 5.222160e-01 5.222515e-01
## 1875 5.311428e-01 5.311781e-01
## 1876 5.400786e-01 5.401138e-01
## 1877 5.490229e-01 5.490580e-01
## 1878 5.579749e-01 5.580100e-01
## 1879 5.669340e-01 5.669690e-01
## 1880 5.758995e-01 5.759343e-01
## 1881 5.848707e-01 5.849054e-01
## 1882 5.938469e-01 5.938815e-01
## 1883 6.028274e-01 6.028619e-01
## 1884 6.118116e-01 6.118459e-01
## 1885 6.207986e-01 6.208328e-01
## 1886 6.297880e-01 6.298220e-01
## 1887 6.387789e-01 6.388127e-01
## 1888 6.477707e-01 6.478044e-01
## 1889 6.567629e-01 6.567965e-01
## 1890 6.657548e-01 6.657882e-01
## 1891 6.747459e-01 6.747791e-01
## 1892 6.837354e-01 6.837684e-01
## 1893 6.927228e-01 6.927556e-01
## 1894 7.017074e-01 7.017401e-01
## 1895 7.106886e-01 7.107212e-01
## 1896 7.196659e-01 7.196983e-01
## 1897 7.286386e-01 7.286708e-01
## 1898 7.376061e-01 7.376380e-01
## 1899 7.465677e-01 7.465995e-01
## 1900 7.555229e-01 7.555545e-01
## 1901 7.644710e-01 7.645024e-01
## 1902 7.734115e-01 7.734426e-01
## 1903 7.823436e-01 7.823745e-01
## 1904 7.912668e-01 7.912975e-01
## 1905 8.001805e-01 8.002110e-01
## 1906 8.090840e-01 8.091143e-01
## 1907 8.179768e-01 8.180068e-01
## 1908 8.268582e-01 8.268880e-01
## 1909 8.357275e-01 8.357571e-01
## 1910 8.445843e-01 8.446136e-01
## 1911 8.534278e-01 8.534569e-01
## 1912 8.622575e-01 8.622863e-01
## 1913 8.710727e-01 8.711013e-01
## 1914 8.798728e-01 8.799011e-01
## 1915 8.886572e-01 8.886852e-01
## 1916 8.974250e-01 8.974528e-01
## 1917 9.061753e-01 9.062029e-01
## 1918 9.149074e-01 9.149346e-01
## 1919 9.236201e-01 9.236472e-01
## 1920 9.323128e-01 9.323395e-01
## 1921 9.409845e-01 9.410109e-01
## 1922 9.496342e-01 9.496604e-01
## 1923 9.582611e-01 9.582870e-01
## 1924 9.668644e-01 9.668900e-01
## 1925 9.754430e-01 9.754683e-01
## 1926 9.839962e-01 9.840212e-01
## 1927 9.925230e-01 9.925477e-01
## 1928 1.001023e+00 1.001047e+00
## 1929 1.009494e+00 1.009518e+00
## 1930 1.017936e+00 1.017960e+00
## 1931 1.026349e+00 1.026372e+00
## 1932 1.034730e+00 1.034753e+00
## 1933 1.043080e+00 1.043103e+00
## 1934 1.051397e+00 1.051420e+00
## 1935 1.059681e+00 1.059703e+00
## 1936 1.067931e+00 1.067952e+00
## 1937 1.076145e+00 1.076166e+00
## 1938 1.084322e+00 1.084344e+00
## 1939 1.092463e+00 1.092484e+00
## 1940 1.100566e+00 1.100587e+00
## 1941 1.108630e+00 1.108650e+00
## 1942 1.116654e+00 1.116674e+00
## 1943 1.124638e+00 1.124657e+00
## 1944 1.132580e+00 1.132599e+00
## 1945 1.140479e+00 1.140498e+00
## 1946 1.148334e+00 1.148353e+00
## 1947 1.156145e+00 1.156163e+00
## 1948 1.163909e+00 1.163927e+00
## 1949 1.171627e+00 1.171644e+00
## 1950 1.179296e+00 1.179313e+00
## 1951 1.186916e+00 1.186933e+00
## 1952 1.194485e+00 1.194502e+00
## 1953 1.202004e+00 1.202020e+00
## 1954 1.209469e+00 1.209485e+00
## 1955 1.216881e+00 1.216896e+00
## 1956 1.224239e+00 1.224253e+00
## 1957 1.231540e+00 1.231555e+00
## 1958 1.238785e+00 1.238799e+00
## 1959 1.245971e+00 1.245985e+00
## 1960 1.253099e+00 1.253112e+00
## 1961 1.260167e+00 1.260179e+00
## 1962 1.267173e+00 1.267185e+00
## 1963 1.274117e+00 1.274129e+00
## 1964 1.280997e+00 1.281009e+00
## 1965 1.287813e+00 1.287825e+00
## 1966 1.294564e+00 1.294575e+00
## 1967 1.301247e+00 1.301258e+00
## 1968 1.307863e+00 1.307873e+00
## 1969 1.314410e+00 1.314420e+00
## 1970 1.320888e+00 1.320897e+00
## 1971 1.327294e+00 1.327303e+00
## 1972 1.333628e+00 1.333636e+00
## 1973 1.339889e+00 1.339897e+00
## 1974 1.346077e+00 1.346084e+00
## 1975 1.352190e+00 1.352197e+00
## 1976 1.358227e+00 1.358234e+00
## 1977 1.364189e+00 1.364195e+00
## 1978 1.370073e+00 1.370079e+00
## 1979 1.375879e+00 1.375885e+00
## 1980 1.381607e+00 1.381612e+00
## 1981 1.387255e+00 1.387260e+00
## 1982 1.392823e+00 1.392827e+00
## 1983 1.398309e+00 1.398313e+00
## 1984 1.403714e+00 1.403717e+00
## 1985 1.409035e+00 1.409038e+00
## 1986 1.414273e+00 1.414275e+00
## 1987 1.419426e+00 1.419428e+00
## 1988 1.424493e+00 1.424495e+00
## 1989 1.429475e+00 1.429476e+00
## 1990 1.434369e+00 1.434370e+00
## 1991 1.439175e+00 1.439176e+00
## 1992 1.443893e+00 1.443893e+00
## 1993 1.448521e+00 1.448520e+00
## 1994 1.453058e+00 1.453058e+00
## 1995 1.457504e+00 1.457503e+00
## 1996 1.461859e+00 1.461857e+00
## 1997 1.466120e+00 1.466118e+00
## 1998 1.470287e+00 1.470285e+00
## 1999 1.474360e+00 1.474357e+00
## 2000 1.478337e+00 1.478334e+00
## 2001 1.482218e+00 1.482214e+00
## 2002 1.486002e+00 1.485998e+00
## 2003 1.489689e+00 1.489685e+00
## 2004 1.493278e+00 1.493273e+00
## 2005 1.496769e+00 1.496763e+00
## 2006 1.500160e+00 1.500154e+00
## 2007 1.503451e+00 1.503444e+00
## 2008 1.506641e+00 1.506634e+00
## 2009 1.509730e+00 1.509723e+00
## 2010 1.512717e+00 1.512710e+00
## 2011 1.515602e+00 1.515594e+00
## 2012 1.518383e+00 1.518375e+00
## 2013 1.521061e+00 1.521052e+00
## 2014 1.523634e+00 1.523625e+00
## 2015 1.526102e+00 1.526092e+00
## 2016 1.528464e+00 1.528454e+00
## 2017 1.530720e+00 1.530709e+00
## 2018 1.532869e+00 1.532858e+00
## 2019 1.534910e+00 1.534898e+00
## 2020 1.536842e+00 1.536831e+00
## 2021 1.538666e+00 1.538654e+00
## 2022 1.540380e+00 1.540368e+00
## 2023 1.541984e+00 1.541971e+00
## 2024 1.543477e+00 1.543464e+00
## 2025 1.544858e+00 1.544844e+00
## 2026 1.546127e+00 1.546113e+00
## 2027 1.547284e+00 1.547269e+00
## 2028 1.548326e+00 1.548312e+00
## 2029 1.549255e+00 1.549240e+00
## 2030 1.550070e+00 1.550055e+00
## 2031 1.550771e+00 1.550755e+00
## 2032 1.551357e+00 1.551340e+00
## 2033 1.551828e+00 1.551811e+00
## 2034 1.552184e+00 1.552167e+00
## 2035 1.552425e+00 1.552407e+00
## 2036 1.552550e+00 1.552532e+00
## 2037 1.552560e+00 1.552542e+00
## 2038 1.552454e+00 1.552435e+00
## 2039 1.552231e+00 1.552212e+00
## 2040 1.551892e+00 1.551872e+00
## 2041 1.551436e+00 1.551416e+00
## 2042 1.550863e+00 1.550843e+00
## 2043 1.550173e+00 1.550152e+00
## 2044 1.549366e+00 1.549345e+00
## 2045 1.548440e+00 1.548419e+00
## 2046 1.547397e+00 1.547376e+00
## 2047 1.546236e+00 1.546214e+00
## 2048 1.544956e+00 1.544934e+00
## 2049 1.543558e+00 1.543535e+00
## 2050 1.542041e+00 1.542018e+00
## 2051 1.540405e+00 1.540381e+00
## 2052 1.538649e+00 1.538625e+00
## 2053 1.536774e+00 1.536750e+00
## 2054 1.534779e+00 1.534754e+00
## 2055 1.532663e+00 1.532639e+00
## 2056 1.530428e+00 1.530403e+00
## 2057 1.528072e+00 1.528047e+00
## 2058 1.525597e+00 1.525571e+00
## 2059 1.523001e+00 1.522975e+00
## 2060 1.520287e+00 1.520260e+00
## 2061 1.517454e+00 1.517427e+00
## 2062 1.514502e+00 1.514475e+00
## 2063 1.511432e+00 1.511405e+00
## 2064 1.508245e+00 1.508217e+00
## 2065 1.504940e+00 1.504912e+00
## 2066 1.501518e+00 1.501490e+00
## 2067 1.497980e+00 1.497951e+00
## 2068 1.494325e+00 1.494296e+00
## 2069 1.490555e+00 1.490525e+00
## 2070 1.486669e+00 1.486639e+00
## 2071 1.482668e+00 1.482638e+00
## 2072 1.478552e+00 1.478522e+00
## 2073 1.474321e+00 1.474291e+00
## 2074 1.469977e+00 1.469947e+00
## 2075 1.465519e+00 1.465489e+00
## 2076 1.460948e+00 1.460917e+00
## 2077 1.456264e+00 1.456233e+00
## 2078 1.451468e+00 1.451436e+00
## 2079 1.446559e+00 1.446527e+00
## 2080 1.441538e+00 1.441506e+00
## 2081 1.436406e+00 1.436374e+00
## 2082 1.431164e+00 1.431131e+00
## 2083 1.425810e+00 1.425777e+00
## 2084 1.420346e+00 1.420313e+00
## 2085 1.414772e+00 1.414739e+00
## 2086 1.409089e+00 1.409056e+00
## 2087 1.403298e+00 1.403264e+00
## 2088 1.397398e+00 1.397365e+00
## 2089 1.391392e+00 1.391358e+00
## 2090 1.385279e+00 1.385245e+00
## 2091 1.379061e+00 1.379026e+00
## 2092 1.372737e+00 1.372703e+00
## 2093 1.366310e+00 1.366275e+00
## 2094 1.359778e+00 1.359743e+00
## 2095 1.353144e+00 1.353109e+00
## 2096 1.346408e+00 1.346372e+00
## 2097 1.339570e+00 1.339534e+00
## 2098 1.332631e+00 1.332596e+00
## 2099 1.325592e+00 1.325557e+00
## 2100 1.318454e+00 1.318418e+00
## 2101 1.311218e+00 1.311182e+00
## 2102 1.303883e+00 1.303847e+00
## 2103 1.296451e+00 1.296414e+00
## 2104 1.288922e+00 1.288886e+00
## 2105 1.281298e+00 1.281261e+00
## 2106 1.273578e+00 1.273541e+00
## 2107 1.265764e+00 1.265727e+00
## 2108 1.257856e+00 1.257819e+00
## 2109 1.249855e+00 1.249818e+00
## 2110 1.241761e+00 1.241724e+00
## 2111 1.233576e+00 1.233539e+00
## 2112 1.225300e+00 1.225263e+00
## 2113 1.216934e+00 1.216897e+00
## 2114 1.208478e+00 1.208441e+00
## 2115 1.199934e+00 1.199896e+00
## 2116 1.191302e+00 1.191264e+00
## 2117 1.182582e+00 1.182545e+00
## 2118 1.173777e+00 1.173739e+00
## 2119 1.164887e+00 1.164849e+00
## 2120 1.155911e+00 1.155874e+00
## 2121 1.146853e+00 1.146815e+00
## 2122 1.137711e+00 1.137673e+00
## 2123 1.128488e+00 1.128450e+00
## 2124 1.119183e+00 1.119145e+00
## 2125 1.109799e+00 1.109761e+00
## 2126 1.100334e+00 1.100296e+00
## 2127 1.090791e+00 1.090753e+00
## 2128 1.081171e+00 1.081132e+00
## 2129 1.071473e+00 1.071435e+00
## 2130 1.061699e+00 1.061661e+00
## 2131 1.051850e+00 1.051812e+00
## 2132 1.041926e+00 1.041888e+00
## 2133 1.031929e+00 1.031891e+00
## 2134 1.021859e+00 1.021821e+00
## 2135 1.011717e+00 1.011679e+00
## 2136 1.001504e+00 1.001466e+00
## 2137 9.912207e-01 9.911827e-01
## 2138 9.808679e-01 9.808298e-01
## 2139 9.704464e-01 9.704084e-01
## 2140 9.599571e-01 9.599191e-01
## 2141 9.494008e-01 9.493630e-01
## 2142 9.387787e-01 9.387409e-01
## 2143 9.280920e-01 9.280542e-01
## 2144 9.173419e-01 9.173042e-01
## 2145 9.065298e-01 9.064921e-01
## 2146 8.956568e-01 8.956192e-01
## 2147 8.847243e-01 8.846868e-01
## 2148 8.737335e-01 8.736960e-01
## 2149 8.626857e-01 8.626483e-01
## 2150 8.515821e-01 8.515448e-01
## 2151 8.404240e-01 8.403868e-01
## 2152 8.292128e-01 8.291757e-01
## 2153 8.179495e-01 8.179125e-01
## 2154 8.066356e-01 8.065987e-01
## 2155 7.952723e-01 7.952355e-01
## 2156 7.838608e-01 7.838242e-01
## 2157 7.724025e-01 7.723659e-01
## 2158 7.608985e-01 7.608621e-01
## 2159 7.493502e-01 7.493139e-01
## 2160 7.377588e-01 7.377226e-01
## 2161 7.261256e-01 7.260896e-01
## 2162 7.144518e-01 7.144159e-01
## 2163 7.027388e-01 7.027030e-01
## 2164 6.909877e-01 6.909521e-01
## 2165 6.791999e-01 6.791644e-01
## 2166 6.673766e-01 6.673413e-01
## 2167 6.555191e-01 6.554839e-01
## 2168 6.436286e-01 6.435936e-01
## 2169 6.317065e-01 6.316716e-01
## 2170 6.197540e-01 6.197193e-01
## 2171 6.077725e-01 6.077380e-01
## 2172 5.957635e-01 5.957292e-01
## 2173 5.837284e-01 5.836943e-01
## 2174 5.716687e-01 5.716347e-01
## 2175 5.595856e-01 5.595519e-01
## 2176 5.474808e-01 5.474473e-01
## 2177 5.353556e-01 5.353222e-01
## 2178 5.232113e-01 5.231782e-01
## 2179 5.110495e-01 5.110166e-01
## 2180 4.988716e-01 4.988389e-01
## 2181 4.866789e-01 4.866464e-01
## 2182 4.744730e-01 4.744407e-01
## 2183 4.622552e-01 4.622231e-01
## 2184 4.500270e-01 4.499951e-01
## 2185 4.377897e-01 4.377581e-01
## 2186 4.255448e-01 4.255134e-01
## 2187 4.132938e-01 4.132626e-01
## 2188 4.010380e-01 4.010071e-01
## 2189 3.887789e-01 3.887482e-01
## 2190 3.765178e-01 3.764874e-01
## 2191 3.642563e-01 3.642261e-01
## 2192 3.519958e-01 3.519658e-01
## 2193 3.397376e-01 3.397078e-01
## 2194 3.274831e-01 3.274537e-01
## 2195 3.152339e-01 3.152047e-01
## 2196 3.029913e-01 3.029624e-01
## 2197 2.907568e-01 2.907281e-01
## 2198 2.785317e-01 2.785033e-01
## 2199 2.663172e-01 2.662890e-01
## 2200 2.541144e-01 2.540865e-01
## 2201 2.419244e-01 2.418968e-01
## 2202 2.297484e-01 2.297211e-01
## 2203 2.175874e-01 2.175604e-01
## 2204 2.054426e-01 2.054159e-01
## 2205 1.933152e-01 1.932887e-01
## 2206 1.812061e-01 1.811799e-01
## 2207 1.691165e-01 1.690906e-01
## 2208 1.570476e-01 1.570220e-01
## 2209 1.450005e-01 1.449752e-01
## 2210 1.329762e-01 1.329512e-01
## 2211 1.209760e-01 1.209512e-01
## 2212 1.090008e-01 1.089764e-01
## 2213 9.705193e-02 9.702778e-02
## 2214 8.513036e-02 8.510652e-02
## 2215 7.323726e-02 7.321372e-02
## 2216 6.137374e-02 6.135051e-02
## 2217 4.954092e-02 4.951800e-02
## 2218 3.773991e-02 3.771730e-02
## 2219 2.597184e-02 2.594954e-02
## 2220 1.423782e-02 1.421583e-02
## 2221 2.538960e-03 2.517287e-03
## 2222 -9.123615e-03 -9.144971e-03
## 2223 -2.074879e-02 -2.076983e-02
## 2224 -3.233545e-02 -3.235617e-02
## 2225 -4.388247e-02 -4.390287e-02
## 2226 -5.538876e-02 -5.540884e-02
## 2227 -6.685344e-02 -6.687319e-02
## 2228 -7.827571e-02 -7.829515e-02
## 2229 -8.965482e-02 -8.967393e-02
## 2230 -1.009900e-01 -1.010088e-01
## 2231 -1.122804e-01 -1.122989e-01
## 2232 -1.235254e-01 -1.235435e-01
## 2233 -1.347241e-01 -1.347419e-01
## 2234 -1.458757e-01 -1.458932e-01
## 2235 -1.569795e-01 -1.569967e-01
## 2236 -1.680348e-01 -1.680516e-01
## 2237 -1.790407e-01 -1.790571e-01
## 2238 -1.899964e-01 -1.900125e-01
## 2239 -2.009013e-01 -2.009171e-01
## 2240 -2.117544e-01 -2.117699e-01
## 2241 -2.225551e-01 -2.225702e-01
## 2242 -2.333026e-01 -2.333174e-01
## 2243 -2.439960e-01 -2.440105e-01
## 2244 -2.546347e-01 -2.546488e-01
## 2245 -2.652178e-01 -2.652316e-01
## 2246 -2.757447e-01 -2.757581e-01
## 2247 -2.862144e-01 -2.862275e-01
## 2248 -2.966262e-01 -2.966390e-01
## 2249 -3.069794e-01 -3.069918e-01
## 2250 -3.172731e-01 -3.172852e-01
## 2251 -3.275067e-01 -3.275184e-01
## 2252 -3.376793e-01 -3.376907e-01
## 2253 -3.477901e-01 -3.478012e-01
## 2254 -3.578385e-01 -3.578492e-01
## 2255 -3.678237e-01 -3.678341e-01
## 2256 -3.777455e-01 -3.777555e-01
## 2257 -3.876034e-01 -3.876131e-01
## 2258 -3.973971e-01 -3.974065e-01
## 2259 -4.071263e-01 -4.071353e-01
## 2260 -4.167905e-01 -4.167992e-01
## 2261 -4.263895e-01 -4.263979e-01
## 2262 -4.359228e-01 -4.359309e-01
## 2263 -4.453902e-01 -4.453979e-01
## 2264 -4.547911e-01 -4.547985e-01
## 2265 -4.641254e-01 -4.641324e-01
## 2266 -4.733925e-01 -4.733992e-01
## 2267 -4.825923e-01 -4.825986e-01
## 2268 -4.917242e-01 -4.917302e-01
## 2269 -5.007880e-01 -5.007937e-01
## 2270 -5.097832e-01 -5.097886e-01
## 2271 -5.187095e-01 -5.187146e-01
## 2272 -5.275666e-01 -5.275714e-01
## 2273 -5.363541e-01 -5.363585e-01
## 2274 -5.450717e-01 -5.450757e-01
## 2275 -5.537189e-01 -5.537226e-01
## 2276 -5.622954e-01 -5.622988e-01
## 2277 -5.708008e-01 -5.708039e-01
## 2278 -5.792349e-01 -5.792376e-01
## 2279 -5.875972e-01 -5.875996e-01
## 2280 -5.958873e-01 -5.958894e-01
## 2281 -6.041050e-01 -6.041068e-01
## 2282 -6.122498e-01 -6.122513e-01
## 2283 -6.203215e-01 -6.203227e-01
## 2284 -6.283204e-01 -6.283213e-01
## 2285 -6.362467e-01 -6.362472e-01
## 2286 -6.441006e-01 -6.441008e-01
## 2287 -6.518824e-01 -6.518823e-01
## 2288 -6.595923e-01 -6.595919e-01
## 2289 -6.672307e-01 -6.672300e-01
## 2290 -6.747977e-01 -6.747967e-01
## 2291 -6.822937e-01 -6.822923e-01
## 2292 -6.897188e-01 -6.897171e-01
## 2293 -6.970733e-01 -6.970714e-01
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## 2295 -7.115718e-01 -7.115692e-01
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## 2298 -7.327965e-01 -7.327930e-01
## 2299 -7.397329e-01 -7.397292e-01
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## 2301 -7.533998e-01 -7.533955e-01
## 2302 -7.601307e-01 -7.601261e-01
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## 2304 -7.733887e-01 -7.733835e-01
## 2305 -7.799163e-01 -7.799108e-01
## 2306 -7.863766e-01 -7.863708e-01
## 2307 -7.927700e-01 -7.927639e-01
## 2308 -7.990966e-01 -7.990902e-01
## 2309 -8.053567e-01 -8.053501e-01
## 2310 -8.115506e-01 -8.115437e-01
## 2311 -8.176785e-01 -8.176714e-01
## 2312 -8.237412e-01 -8.237337e-01
## 2313 -8.297393e-01 -8.297316e-01
## 2314 -8.356736e-01 -8.356656e-01
## 2315 -8.415449e-01 -8.415366e-01
## 2316 -8.473538e-01 -8.473453e-01
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## 2318 -8.587878e-01 -8.587788e-01
## 2319 -8.644144e-01 -8.644051e-01
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## 2321 -8.754903e-01 -8.754805e-01
## 2322 -8.809412e-01 -8.809311e-01
## 2323 -8.863350e-01 -8.863247e-01
## 2324 -8.916726e-01 -8.916619e-01
## 2325 -8.969545e-01 -8.969437e-01
## 2326 -9.021817e-01 -9.021706e-01
## 2327 -9.073548e-01 -9.073434e-01
## 2328 -9.124746e-01 -9.124629e-01
## 2329 -9.175418e-01 -9.175299e-01
## 2330 -9.225572e-01 -9.225451e-01
## 2331 -9.275216e-01 -9.275092e-01
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## 2334 -9.421158e-01 -9.421027e-01
## 2335 -9.468834e-01 -9.468700e-01
## 2336 -9.516036e-01 -9.515901e-01
## 2337 -9.562774e-01 -9.562636e-01
## 2338 -9.609053e-01 -9.608912e-01
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## 2345 -9.920872e-01 -9.920715e-01
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## 2349 -1.009030e+00 -1.009013e+00
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## 19835 -4.048105e-01 -4.048074e-01
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## 19837 -3.979976e-01 -3.979927e-01
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## 19853 -3.423170e-01 -3.422972e-01
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## 19855 -3.351706e-01 -3.351489e-01
## 19856 -3.315787e-01 -3.315560e-01
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## 19858 -3.243560e-01 -3.243315e-01
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## 19865 -2.986385e-01 -2.986072e-01
## 19866 -2.949054e-01 -2.948732e-01
## 19867 -2.911568e-01 -2.911237e-01
## 19868 -2.873924e-01 -2.873583e-01
## 19869 -2.836118e-01 -2.835768e-01
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## 19872 -2.721713e-01 -2.721334e-01
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## 19875 -2.605768e-01 -2.605361e-01
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## 19878 -2.488214e-01 -2.487778e-01
## 19879 -2.448659e-01 -2.448214e-01
## 19880 -2.408915e-01 -2.408461e-01
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## 19883 -2.288523e-01 -2.288041e-01
## 19884 -2.247997e-01 -2.247505e-01
## 19885 -2.207269e-01 -2.206768e-01
## 19886 -2.166336e-01 -2.165826e-01
## 19887 -2.125196e-01 -2.124676e-01
## 19888 -2.083847e-01 -2.083318e-01
## 19889 -2.042286e-01 -2.041748e-01
## 19890 -2.000512e-01 -1.999965e-01
## 19891 -1.958523e-01 -1.957967e-01
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## 19893 -1.873891e-01 -1.873317e-01
## 19894 -1.831244e-01 -1.830661e-01
## 19895 -1.788373e-01 -1.787781e-01
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## 19898 -1.658400e-01 -1.657781e-01
## 19899 -1.614615e-01 -1.613988e-01
## 19900 -1.570597e-01 -1.569961e-01
## 19901 -1.526343e-01 -1.525698e-01
## 19902 -1.481851e-01 -1.481197e-01
## 19903 -1.437120e-01 -1.436457e-01
## 19904 -1.392147e-01 -1.391476e-01
## 19905 -1.346930e-01 -1.346250e-01
## 19906 -1.301467e-01 -1.300779e-01
## 19907 -1.255756e-01 -1.255060e-01
## 19908 -1.209796e-01 -1.209091e-01
## 19909 -1.163584e-01 -1.162870e-01
## 19910 -1.117117e-01 -1.116395e-01
## 19911 -1.070395e-01 -1.069665e-01
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## 19913 -9.761740e-02 -9.754275e-02
## 19914 -9.286715e-02 -9.279169e-02
## 19915 -8.809050e-02 -8.801424e-02
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## 19918 -7.360084e-02 -7.352221e-02
## 19919 -6.871734e-02 -6.863792e-02
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## 19921 -5.886922e-02 -5.878826e-02
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## 19926 -3.377036e-02 -3.368565e-02
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## 19935 1.316914e-02 1.326014e-02
## 19936 1.852733e-02 1.861898e-02
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## 19938 2.933069e-02 2.942365e-02
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## 20100 1.091423e+00 1.091472e+00
## 20101 1.096997e+00 1.097045e+00
## 20102 1.102528e+00 1.102574e+00
## 20103 1.108015e+00 1.108060e+00
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## 20105 1.118857e+00 1.118900e+00
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## 20117 1.179942e+00 1.179971e+00
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## 20122 1.203218e+00 1.203242e+00
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## 32005 2.502650e+00 2.504823e+00
## 32006 2.516681e+00 2.518856e+00
## 32007 2.530617e+00 2.532792e+00
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## 40898 7.403450e-02 7.291040e-02
## 40899 1.052297e-01 1.041102e-01
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## 40901 1.676695e-01 1.665595e-01
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## 40903 2.301525e-01 2.290522e-01
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## 40907 3.551398e-01 3.540597e-01
## 40908 3.863713e-01 3.852963e-01
## 40909 4.175898e-01 4.165201e-01
## 40910 4.487920e-01 4.477276e-01
## 40911 4.799745e-01 4.789154e-01
## 40912 5.111338e-01 5.100802e-01
## 40913 5.422666e-01 5.412185e-01
## 40914 5.733696e-01 5.723270e-01
## 40915 6.044392e-01 6.034022e-01
## 40916 6.354719e-01 6.344406e-01
## 40917 6.664642e-01 6.654386e-01
## 40918 6.974123e-01 6.963925e-01
## 40919 7.283128e-01 7.272988e-01
## 40920 7.591619e-01 7.581538e-01
## 40921 7.899561e-01 7.889539e-01
## 40922 8.206917e-01 8.196955e-01
## 40923 8.513652e-01 8.503751e-01
## 40924 8.819729e-01 8.809889e-01
## 40925 9.125113e-01 9.115333e-01
## 40926 9.429766e-01 9.420049e-01
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## 40932 1.124034e+00 1.123100e+00
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## 40934 1.183638e+00 1.182718e+00
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## 40936 1.242821e+00 1.241914e+00
## 40937 1.272245e+00 1.271345e+00
## 40938 1.301553e+00 1.300659e+00
## 40939 1.330741e+00 1.329854e+00
## 40940 1.359805e+00 1.358925e+00
## 40941 1.388742e+00 1.387869e+00
## 40942 1.417548e+00 1.416682e+00
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## 40944 1.474754e+00 1.473902e+00
## 40945 1.503146e+00 1.502301e+00
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## 40960 1.909581e+00 1.908847e+00
## 40961 1.935217e+00 1.934490e+00
## 40962 1.960649e+00 1.959929e+00
## 40963 1.985874e+00 1.985163e+00
## 40964 2.010890e+00 2.010186e+00
## 40965 2.035691e+00 2.034995e+00
## 40966 2.060275e+00 2.059587e+00
## 40967 2.084638e+00 2.083958e+00
## 40968 2.108777e+00 2.108104e+00
## 40969 2.132687e+00 2.132022e+00
## 40970 2.156365e+00 2.155708e+00
## 40971 2.179807e+00 2.179158e+00
## 40972 2.203011e+00 2.202370e+00
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## 40975 2.271158e+00 2.270541e+00
## 40976 2.293375e+00 2.292767e+00
## 40977 2.315339e+00 2.314739e+00
## 40978 2.337046e+00 2.336453e+00
## 40979 2.358493e+00 2.357908e+00
## 40980 2.379677e+00 2.379101e+00
## 40981 2.400595e+00 2.400027e+00
## 40982 2.421244e+00 2.420684e+00
## 40983 2.441621e+00 2.441069e+00
## 40984 2.461723e+00 2.461180e+00
## 40985 2.481547e+00 2.481012e+00
## 40986 2.501091e+00 2.500564e+00
## 40987 2.520350e+00 2.519832e+00
## 40988 2.539323e+00 2.538813e+00
## 40989 2.558005e+00 2.557503e+00
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## 49791 1.001320e+00 1.001281e+00
## 49792 1.002978e+00 1.002940e+00
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## 49794 1.006288e+00 1.006251e+00
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## 49796 1.009585e+00 1.009548e+00
## 49797 1.011179e+00 1.011143e+00
## 49798 1.012674e+00 1.012639e+00
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## 49803 -1.303633e+00 -1.303558e+00
## 49804 -1.305884e+00 -1.305810e+00
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## 49806 -1.309846e+00 -1.309774e+00
## 49807 -1.311557e+00 -1.311485e+00
## 49808 -1.313087e+00 -1.313015e+00
## 49809 -1.314434e+00 -1.314363e+00
## 49810 -1.315600e+00 -1.315530e+00
## 49811 -1.316583e+00 -1.316513e+00
## 49812 -1.317382e+00 -1.317313e+00
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## 49814 -1.318428e+00 -1.318360e+00
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## 49816 -1.318734e+00 -1.318667e+00
## 49817 -1.318608e+00 -1.318542e+00
## 49818 -1.318295e+00 -1.318230e+00
## 49819 -1.317795e+00 -1.317730e+00
## 49820 -1.317107e+00 -1.317043e+00
## 49821 -1.316231e+00 -1.316167e+00
## 49822 -1.315166e+00 -1.315103e+00
## 49823 -1.313911e+00 -1.313849e+00
## 49824 -1.312466e+00 -1.312405e+00
## 49825 -1.310831e+00 -1.310770e+00
## 49826 -1.309004e+00 -1.308944e+00
## 49827 -1.306986e+00 -1.306927e+00
## 49828 -1.304776e+00 -1.304717e+00
## 49829 -1.302372e+00 -1.302314e+00
## 49830 -1.299775e+00 -1.299718e+00
## 49831 -1.296985e+00 -1.296929e+00
## 49832 -1.294002e+00 -1.293946e+00
## 49833 -1.290826e+00 -1.290771e+00
## 49834 -1.287457e+00 -1.287402e+00
## 49835 -1.283895e+00 -1.283842e+00
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## 49837 -1.276195e+00 -1.276143e+00
## 49838 -1.272057e+00 -1.272006e+00
## 49839 -1.267728e+00 -1.267677e+00
## 49840 -1.263207e+00 -1.263157e+00
## 49841 -1.258494e+00 -1.258445e+00
## 49842 -1.253591e+00 -1.253542e+00
## 49843 -1.248496e+00 -1.248449e+00
## 49844 -1.243211e+00 -1.243164e+00
## 49845 -1.237736e+00 -1.237690e+00
## 49846 -1.232070e+00 -1.232025e+00
## 49847 -1.226214e+00 -1.226170e+00
## 49848 -1.220169e+00 -1.220125e+00
## 49849 -1.213934e+00 -1.213891e+00
## 49850 -1.207509e+00 -1.207467e+00
## 49851 -1.200896e+00 -1.200854e+00
## 49852 -1.194093e+00 -1.194053e+00
## 49853 -1.187102e+00 -1.187062e+00
## 49854 -1.179922e+00 -1.179883e+00
## 49855 -1.172553e+00 -1.172515e+00
## 49856 -1.164997e+00 -1.164960e+00
## 49857 -1.157253e+00 -1.157216e+00
## 49858 -1.149320e+00 -1.149285e+00
## 49859 -1.141201e+00 -1.141167e+00
## 49860 -1.132896e+00 -1.132862e+00
## 49861 -1.124406e+00 -1.124373e+00
## 49862 -1.115732e+00 -1.115700e+00
## 49863 -1.106875e+00 -1.106844e+00
## 49864 -1.097837e+00 -1.097807e+00
## 49865 -1.088618e+00 -1.088589e+00
## 49866 -1.079220e+00 -1.079191e+00
## 49867 -1.069643e+00 -1.069615e+00
## 49868 -1.059888e+00 -1.059861e+00
## 49869 -1.049958e+00 -1.049932e+00
## 49870 -1.039852e+00 -1.039827e+00
## 49871 -1.029572e+00 -1.029548e+00
## 49872 -1.019120e+00 -1.019096e+00
## 49873 -1.008495e+00 -1.008473e+00
## 49874 -9.976996e-01 -9.976781e-01
## 49875 -9.867344e-01 -9.867138e-01
## 49876 -9.756006e-01 -9.755809e-01
## 49877 -9.642993e-01 -9.642805e-01
## 49878 -9.528315e-01 -9.528136e-01
## 49879 -9.411983e-01 -9.411813e-01
## 49880 -9.294008e-01 -9.293847e-01
## 49881 -9.174402e-01 -9.174250e-01
## 49882 -9.053174e-01 -9.053031e-01
## 49883 -8.930336e-01 -8.930202e-01
## 49884 -8.805899e-01 -8.805774e-01
## 49885 -8.679873e-01 -8.679757e-01
## 49886 -8.552269e-01 -8.552163e-01
## 49887 -8.423100e-01 -8.423003e-01
## 49888 -8.292380e-01 -8.292292e-01
## 49889 -8.160126e-01 -8.160047e-01
## 49890 -8.026351e-01 -8.026281e-01
## 49891 -7.891073e-01 -7.891012e-01
## 49892 -7.754306e-01 -7.754254e-01
## 49893 -7.616066e-01 -7.616024e-01
## 49894 -7.476368e-01 -7.476335e-01
## 49895 -7.335228e-01 -7.335205e-01
## 49896 -7.192662e-01 -7.192648e-01
## 49897 -7.048685e-01 -7.048680e-01
## 49898 -6.903312e-01 -6.903316e-01
## 49899 -6.756559e-01 -6.756573e-01
## 49900 -6.608442e-01 -6.608465e-01
## 49901 -6.458975e-01 -6.459008e-01
## 49902 -6.308176e-01 -6.308217e-01
## 49903 -6.156058e-01 -6.156109e-01
## 49904 -6.002637e-01 -6.002698e-01
## 49905 -5.847930e-01 -5.848000e-01
## 49906 -5.691951e-01 -5.692030e-01
## 49907 -5.534716e-01 -5.534805e-01
## 49908 -5.376241e-01 -5.376339e-01
## 49909 -5.216540e-01 -5.216648e-01
## 49910 -5.055631e-01 -5.055747e-01
## 49911 -4.893527e-01 -4.893653e-01
## 49912 -4.730245e-01 -4.730380e-01
## 49913 -4.565800e-01 -4.565944e-01
## 49914 -4.400207e-01 -4.400361e-01
## 49915 -4.233483e-01 -4.233647e-01
## 49916 -4.065647e-01 -4.065820e-01
## 49917 -3.896721e-01 -3.896903e-01
## 49918 -3.726724e-01 -3.726916e-01
## 49919 -3.555679e-01 -3.555880e-01
## 49920 -3.383606e-01 -3.383817e-01
## 49921 -3.210526e-01 -3.210746e-01
## 49922 -3.036460e-01 -3.036689e-01
## 49923 -2.861429e-01 -2.861668e-01
## 49924 -2.685454e-01 -2.685702e-01
## 49925 -2.508556e-01 -2.508813e-01
## 49926 -2.330755e-01 -2.331022e-01
## 49927 -2.152074e-01 -2.152350e-01
## 49928 -1.972532e-01 -1.972817e-01
## 49929 -1.792151e-01 -1.792445e-01
## 49930 -1.610952e-01 -1.611255e-01
## 49931 -1.428955e-01 -1.429268e-01
## 49932 -1.246182e-01 -1.246504e-01
## 49933 -1.062654e-01 -1.062985e-01
## 49934 -8.783913e-02 -8.787317e-02
## 49935 -6.934150e-02 -6.937646e-02
## 49936 -5.077462e-02 -5.081050e-02
## 49937 -3.214059e-02 -3.217738e-02
## 49938 -1.344150e-02 -1.347920e-02
## 49939 5.320558e-03 5.281949e-03
## 49940 2.414348e-02 2.410396e-02
## 49941 4.302517e-02 4.298475e-02
## 49942 6.196354e-02 6.192221e-02
## 49943 8.095647e-02 8.091425e-02
## 49944 1.000017e-01 9.995861e-02
## 49945 1.190970e-01 1.190529e-01
## 49946 1.382398e-01 1.381949e-01
## 49947 1.574278e-01 1.573820e-01
## 49948 1.766588e-01 1.766121e-01
## 49949 1.959302e-01 1.958826e-01
## 49950 2.152398e-01 2.151913e-01
## 49951 2.345851e-01 2.345358e-01
## 49952 2.539639e-01 2.539137e-01
## 49953 2.733737e-01 2.733226e-01
## 49954 2.928122e-01 2.927602e-01
## 49955 3.122770e-01 3.122242e-01
## 49956 3.317658e-01 3.317121e-01
## 49957 3.512761e-01 3.512216e-01
## 49958 3.708057e-01 3.707503e-01
## 49959 3.903521e-01 3.902959e-01
## 49960 4.099131e-01 4.098560e-01
## 49961 4.294861e-01 4.294282e-01
## 49962 4.490689e-01 4.490102e-01
## 49963 4.686591e-01 4.685995e-01
## 49964 4.882544e-01 4.881940e-01
## 49965 5.078523e-01 5.077911e-01
## 49966 5.274506e-01 5.273885e-01
## 49967 5.470467e-01 5.469838e-01
## 49968 5.666385e-01 5.665748e-01
## 49969 5.862234e-01 5.861589e-01
## 49970 6.057992e-01 6.057339e-01
## 49971 6.253635e-01 6.252974e-01
## 49972 6.449138e-01 6.448469e-01
## 49973 6.644474e-01 6.643797e-01
## 49974 6.839617e-01 6.838932e-01
## 49975 7.034539e-01 7.033847e-01
## 49976 7.229214e-01 7.228514e-01
## 49977 7.423615e-01 7.422907e-01
## 49978 7.617714e-01 7.616998e-01
## 49979 7.811486e-01 7.810762e-01
## 49980 8.004902e-01 8.004171e-01
## 49981 8.197937e-01 8.197198e-01
## 49982 8.390563e-01 8.389817e-01
## 49983 8.582753e-01 8.582000e-01
## 49984 8.774480e-01 8.773720e-01
## 49985 8.965719e-01 8.964951e-01
## 49986 9.156441e-01 9.155666e-01
## 49987 9.346619e-01 9.345837e-01
## 49988 9.536228e-01 9.535439e-01
## 49989 9.725239e-01 9.724443e-01
## 49990 9.913627e-01 9.912824e-01
## 49991 1.010136e+00 1.010055e+00
## 49992 1.028842e+00 1.028761e+00
## 49993 1.047478e+00 1.047395e+00
## 49994 1.066040e+00 1.065957e+00
## 49995 1.084526e+00 1.084443e+00
## 49996 1.102934e+00 1.102850e+00
## 49997 1.121261e+00 1.121176e+00
## 49998 1.139504e+00 1.139418e+00
## 49999 1.157660e+00 1.157574e+00
## [ reached 'max' / getOption("max.print") -- omitted 70001 rows ]4 Reference method
4.1 GLMMadaptive
- Here we can fit a model with random intercept and slope for time
\[g(E(Y_i(t))) = \beta_0+\beta_1t+b_{i0}+b_{i1}t\] - But this model is obviously too simple for the data generation scheme
# model fit
t1 <- Sys.time()
fit_adglmm <- mixed_model(Y ~ t, random = ~ t | id, data = train_df, family = binomial())
t2 <- Sys.time()
# time
t2-t1## Time difference of 2.23976 mins# model
summary(fit_adglmm)##
## Call:
## mixed_model(fixed = Y ~ t, random = ~t | id, data = train_df,
## family = binomial())
##
## Data Descriptives:
## Number of Observations: 500000
## Number of Groups: 500
##
## Model:
## family: binomial
## link: logit
##
## Fit statistics:
## log.Lik AIC BIC
## -324123.9 648257.7 648278.8
##
## Random effects covariance matrix:
## StdDev Corr
## (Intercept) 1.1840
## t 2.3831 -0.9987
##
## Fixed effects:
## Estimate Std.Err z-value p-value
## (Intercept) 0.0615 0.0533 1.1546 0.24827
## t -0.1185 0.1071 -1.1059 0.26877
##
## Integration:
## method: adaptive Gauss-Hermite quadrature rule
## quadrature points: 11
##
## Optimization:
## method: EM
## converged: TRUE4.2 Flexible GLMM
- Fit on all 500 subjects take too long. I reduced the sample size to 10
- In fact, 10 samples took more than 30 minutes to fit the model. Therefore, I further reduce the number of observations for each subject. I’ll start with only the 1/10 of the time track (100 observations), take observations at bin midpoints 5, 15, 25, …., 995. This is to preserve the non-linearity of latent function when reducing grid density
\[g(E(Y_i(t))) = \sum_{k=1}^5\zeta_{k}B_k(t)+\sum_{l=1}^4\xi_{il}\phi_l(t)\]
# subset 10 subjects from training dataset
df_train_sub <- train_df %>% filter(id %in% 1:10 & sind==bin)
df_train_sub$id <- droplevels(df_train_sub$id)
# levels(df_train_sub$id)
# intersect(t, unique(df_train_sub$t))- Below I construct basis functions for the fixed effect/random intercept. Let’s use five basis functions to start with. I believe this is far less flexible than the FPCA model, but it still takes a really long time to run. It is actually not realistic to repeat 500 times.
PS 50 subjects took 1.06 hours to fit with the same model structure.
# construct design matrix for the fixed effect intercept
t_sub <- unique(df_train_sub$t)
fix_xmat <- smoothCon(s(t, bs="cr", k=5), data = data.frame(t=t_sub))
dim(fix_xmat[[1]]$X)## [1] 100 5# basis functions
fix_xmat <- data.frame(t=t_sub,
fix_xmat[[1]]$X)
colnames(fix_xmat) <- c("t", paste0("B", 1:5))
df_train_sub <- df_train_sub %>% left_join(fix_xmat, by = "t")- Below I construct basis functions for the individual random effect (also a functinal intercept), with four basis functions to start with.
# construct design matrix for the random effect
rand_xmat <- smoothCon(s(t, bs="cr", k=4), data = data.frame(t=t_sub))
dim(rand_xmat[[1]]$X) # 5 basis functions for each of the 50 subjects## [1] 100 4rand_xmat <- data.frame(t = t, rand_xmat[[1]]$X)
df_train_sub <- df_train_sub %>%
left_join(rand_xmat, by = c("t"))- Now I’ll fit the model and record the time spent on model fitting
The model fitting took about 6 minutes at this data scale, which obviously is not realistic for 500 simulated datasets.
Also tried increasing number of basis functions, which ended up with “computationally singular”. In fact, this method does not seem very numerically stable.
# model fit
t1 <- Sys.time()
fit_adglmm <- mixed_model(fixed = Y ~ 0+B1+B2+B3+B4+B5,
random = ~X1+X2+X3+X4 | id,
data = df_train_sub, family = binomial())
t2 <- Sys.time()
t2-t1## Time difference of 5.761589 mins# model
summary(fit_adglmm)##
## Call:
## mixed_model(fixed = Y ~ 0 + B1 + B2 + B3 + B4 + B5, random = ~X1 +
## X2 + X3 + X4 | id, data = df_train_sub, family = binomial())
##
## Data Descriptives:
## Number of Observations: 1000
## Number of Groups: 10
##
## Model:
## family: binomial
## link: logit
##
## Fit statistics:
## log.Lik AIC BIC
## -692.0652 1424.13 1430.182
##
## Random effects covariance matrix:
## StdDev Corr
## (Intercept) 0.4447 (Intr) X1 X2 X3
## X1 0.4525 -0.9789
## X2 0.4473 -0.9889 0.9535
## X3 0.4468 -0.9906 0.9800 0.9755
## X4 0.4603 -0.9722 0.9478 0.9499 0.9514
##
## Fixed effects:
## Estimate Std.Err z-value p-value
## B1 0.3231 0.2490 1.2977 0.19441
## B2 -0.0915 0.1407 -0.6505 0.51534
## B3 -0.0326 0.1397 -0.2337 0.81520
## B4 -0.0570 0.1406 -0.4052 0.68530
## B5 0.1543 0.2476 0.6230 0.53326
##
## Integration:
## method: adaptive Gauss-Hermite quadrature rule
## quadrature points: 7
##
## Optimization:
## method: hybrid EM and quasi-Newton
## converged: TRUE- But anyways, let’s try to use this method for prediction. Let’s do that for 10 new test subjects.
df_test_sub <- test_df %>% filter(id %in% 1:10 & sind==bin)
df_test_sub$id <- droplevels(df_test_sub$id)
# levels(df_test_sub$id)
# basis functions for both fixed and random effects should be shared across sample
df_test_sub <- df_test_sub %>% left_join(fix_xmat, by = "t") %>%
left_join(rand_xmat, by = "t")
# given observation from 0.2, 0.4, 0.6, 0.8, predict the rest
# this step in fact is very fast
pred_adglmm1 <- predict(fit_adglmm,
newdata = df_test_sub %>% filter(t<=0.2),
newdata2 = df_test_sub %>% filter(t>0.2),
type = "subject_specific", type_pred = "link",
return_newdata = TRUE)
pred_adglmm2 <- predict(fit_adglmm,
newdata = df_test_sub %>% filter(t<=0.4),
newdata2 = df_test_sub %>% filter(t>0.4),
type = "subject_specific", type_pred = "link",
return_newdata = TRUE)
pred_adglmm3 <- predict(fit_adglmm,
newdata = df_test_sub %>% filter(t<=0.6),
newdata2 = df_test_sub %>% filter(t>0.6),
type = "subject_specific", type_pred = "link",
return_newdata = TRUE)
pred_adglmm4 <- predict(fit_adglmm,
newdata = df_test_sub %>% filter(t<=0.8),
newdata2 = df_test_sub %>% filter(t>0.8),
type = "subject_specific", type_pred = "link",
return_newdata = TRUE)pred_adglmm_all <- df_test_sub %>%
left_join(pred_adglmm1$newdata2 %>% select(id, t, pred), by = c("id", "t")) %>%
left_join(pred_adglmm2$newdata2 %>% select(id, t, pred), by = c("id", "t")) %>%
left_join(pred_adglmm3$newdata2 %>% select(id, t, pred), by = c("id", "t")) %>%
left_join(pred_adglmm4$newdata2 %>% select(id, t, pred), by = c("id", "t"))
colnames(pred_adglmm_all) <- c(colnames(pred_adglmm_all)[1:15], "pred1", "pred2", "pred3", "pred4")pred_adglmm_all %>%
filter(id %in% 1:4) %>%
mutate_at(vars(eta_i, pred1, pred2, pred3, pred4), function(x){exp(x)/(1+exp(x))}) %>%
ggplot()+
geom_point(aes(x=t, y=Y), size = 0.2)+
geom_line(aes(x=t, y=eta_i, col = "True"))+
geom_line(aes(x=t, y=pred1, col = "0.2"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred2, col = "0.4"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred3, col = "0.5"), linetype="dashed", na.rm=T)+
geom_line(aes(x=t, y=pred4, col = "0.8"), linetype="dashed", na.rm=T)+
facet_wrap(~id)+
labs(title = "GLMMadaptive + spline basis prediction")